From Geometric Quantization to Conformal Field Theory

Abstract

In this paper we proceed with studying the 2d conformal field theories in terms of geometric quantization (see [1,2,3]). As it was shown in our preceding papers the standard geometric quantization method [4] can be reformulated in terms of the path integral approach. In Ref. [2] the correspondence between the coadjoint orbit and the irreducible representation of compact Lie groups was explicitly realized by means of the functional integral. More precisely, we constructed in [2] a quantum mechanical system, such one that the path integral with boundary conditions gives matrix coefficients of the relevant irreducible representation. The action functional of this system is defined by the canonical symplectic structure Ω on the given coadjoint orbit and a Hamiltonian H (X), which is a function on the orbit \( :s = \int {{d^{ - 1}}\Omega - \int {H(X)} } dt \); this action is a functional of trajectories on the orbit. Further in [3] using the same rules we described quantum field theory on the coadjoint orbit of infinitedimensional Lie groups (Virasoro, Kac-Moody) and studied the properties of the relevant action functional. In particular, we have shown that for the Virasoro group the geometrical action, written in terms of group variables F(x)∈ diff S 1 differs from the action in 2d gravity by an extra term \( \int {{b_0}\dot FF'dxdt} \), where the number b 0 parameterizes generic coadjoint orbits.KeywordsModel SpaceConformal Field TheoryGeometric QuantizationWeyl ChamberCoadjoint OrbitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.